Question

Find the domain and range of the following real functions:

(i) $f\left(x\right)=-|x|$ (ii) $f\left(x\right)=\sqrt{9-x^2}$

Answer 1

(i) Here f(x) = - |x|

The function is defined for all real values of x.

$\therefore$ Domain of function = R

Now when x < 0, |x| = - x

$\therefore f\left(x\right)=-(-x)=x<0$

When x = 0, |x| = 0

$\therefore f\left(x\right)=-x<0$

So $f\left(x\right)\le 0$ for all real values of x.

$\therefore$ Range of function $=(-\infty ,0].$

(ii) Here $f\left(x\right)=\sqrt{9-x^2}$

The function is not defined when $9-x^2<0.$

$\therefore$ Domain of function $=\{x:9-x^2\ge 0\}$

$=\{x:x^2-9\le 0\}$

$=\{x:(x+3\left)\right(x-3)\le 0\}=[-3,3]$

Now $f\left(x\right)=\sqrt{9-x^2}\ge 0$ for $xϵ[-3,3]$

$\therefore$ Range of function $=[0,\infty ).$